Problem: $ E = \left[\begin{array}{rr}4 & 3 \\ 3 & 1 \\ 4 & 2\end{array}\right]$ $ F = \left[\begin{array}{rr}-1 & -1 \\ -1 & 4\end{array}\right]$ What is $ E F$ ?
Solution: Because $ E$ has dimensions $(3\times2)$ and $ F$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(3\times2)$ $ E F = \left[\begin{array}{rr}{4} & {3} \\ {3} & {1} \\ \color{gray}{4} & \color{gray}{2}\end{array}\right] \left[\begin{array}{rr}{-1} & \color{#DF0030}{-1} \\ {-1} & \color{#DF0030}{4}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ E$ , with the corresponding elements in column $j$ of the second matrix, $ F$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ E$ with the first element in ${\text{column }1}$ of $ F$ , then multiply the second element in ${\text{row }1}$ of $ E$ with the second element in ${\text{column }1}$ of $ F$ , and so on. Add the products together. $ \left[\begin{array}{rr}{4}\cdot{-1}+{3}\cdot{-1} & ? \\ ? & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ E$ with the corresponding elements in ${\text{column }1}$ of $ F$ and add the products together. $ \left[\begin{array}{rr}{4}\cdot{-1}+{3}\cdot{-1} & ? \\ {3}\cdot{-1}+{1}\cdot{-1} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ E$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ F$ and add the products together. $ \left[\begin{array}{rr}{4}\cdot{-1}+{3}\cdot{-1} & {4}\cdot\color{#DF0030}{-1}+{3}\cdot\color{#DF0030}{4} \\ {3}\cdot{-1}+{1}\cdot{-1} & ? \\ ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{4}\cdot{-1}+{3}\cdot{-1} & {4}\cdot\color{#DF0030}{-1}+{3}\cdot\color{#DF0030}{4} \\ {3}\cdot{-1}+{1}\cdot{-1} & {3}\cdot\color{#DF0030}{-1}+{1}\cdot\color{#DF0030}{4} \\ \color{gray}{4}\cdot{-1}+\color{gray}{2}\cdot{-1} & \color{gray}{4}\cdot\color{#DF0030}{-1}+\color{gray}{2}\cdot\color{#DF0030}{4}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}-7 & 8 \\ -4 & 1 \\ -6 & 4\end{array}\right] $